初中《数学》第四册简介

初中《数学》第四册(试用本)共有三章。 前两章是代数内容。其中“一元二次方程”一章包含了一元二次方程的解法、根的判别式及根与系数的关系,还包含了能归于一元二次方程解法的一些高次方程、分式方程、根式方程以及一些二元二次方程组的解法。“指数和常用对数”一章包含了零指数、负     

解三角题应注意的一些误区

在三角函数这一章里 ,由于公式多 ,因而解题方法比较灵活 ,如果解法选择不当 ,不仅运算麻烦 ,而且有时还会改变解集 .由于三角函数的独特性质 ,解题时若注意不到或挖掘不彻底 ,也会陷入不可自拔的误区中 .本文通过举例 ,来说明这种现象 .1　误区之一　解法不当引起复杂的运算有些三角问题 ,若解法不当 ,就需分类讨论 ,运算量大 ,易出错 ,若选择恰当的解法 ,则可避免解题过程复杂化 .例 1　若ｓｉｎ θ2 =35 ,ｃｏｓ θ2 =- 45 ,判断θ是第几象限的角 .解法 1　∵ｓｉｎ θ2 =35 >12 ,∴ 2ｋπ +π6<θ2 <2ｋπ +5π6　　 (ｋ∈Ｚ) .即 4ｋπ…     

在求异中创新

今年开始 ,全日制普通高级中学 (试验修订本·必修 )《数学》课本 (简称新教材 )已在全国 2 0多个省市自治区使用 ,同时新大纲也开始实施 .“培养学生的创新精神”是新大纲不同于原大纲的明显变化之一 .作为高中学生 ,可以在平时的阅读和解题中逐步培养自己的创新精神 ,善于从平凡的解法中发现新奇、简洁或其它不同的解法 ,标新立异 .现举新教材第三章《数列》中的几例供同学们参考 .1　“求异”在多种解法中例 1　等比数列求和公式的推导 (新教材第 1 30页 ) .教材上的推导 ,我们一般称“错项相减法” ,除此之外 ,我们再给出两种推导公式的…     

几种三角函数的最值问题

三角函数的最值问题涉及范围广,方法典型独特,解法多样,有些解法又有较强的技巧性,是三角函数一章学习中的重点和难点,下面介绍几种常见的题型及解法。1.对于形如y=asinx b或y=acosx b(a≠0)的三角函数的最值问题,可从中解出     

异面直线所成角问题的多种解法

异面直线所成的角是必修2第二章第一节《空间点、直线、平面之间的位置关系》中的内容,也是高考的考点之一,多以选择题或解答题为主的形式考查,多为中档题.在高三的一轮复习中,这部分内容被安排在了第七章中,本人以一轮复习资料《创新大课堂》中的本部分的一道题为例来浅析两条异面直线所成角的解法.     

异面直线所成角问题的多种解法

<正>异面直线所成的角是必修2第二章第一节《空间点、直线、平面之间的位置关系》中的内容,也是高考的考点之一,多以选择题或解答题为主的形式考查,多为中档题.在高三的一轮复习中,这部分内容被安排在了第七章中,本人以一轮复习资料《创新大课堂》中的本部分的一道题为例来浅析两条异面直线所成角的解法.     

一道课本习题的多解法探讨

沪科版八年级数学教材在平行四边形一章中有这样一道试题:求证梯形两条对角线中点的连线平行两底,且等于两底差的一半.这道题也有多种解法.分析这是一道文字叙述式的证明题.其解法步骤:先画出符合题意的图形,再写出已     

四类对称问题的解法

解析几何第一章内容的对称问题在高考中频频出现.而对称问题由于解法多,同学们难以择其善者而从之,故就显得较难.以下是笔者对对称问题的归类及解法分析. 对称问题大致上有以下四类:点关于点对称;点关于线对称;线关于点对称;线关于线对称.     

对一节课中例题设置的商榷及建议

笔者最近教授上教版高中数学第八章第三节“平面向量的分解定理”一课时,发现例题的设置及解法会影响本节课目标的达成.尽管教参与考试手册中都提到了理解平面向量分解定理这一目标,但学生普遍感到困惑:不学习这节课的内容,这3道例题也可以很好地完成,那为什么还要学习分解定理?这3道例题存在的意义何在?     

“九章算术”中关於“方程”解法的成就

一九章算术是我国最古老的算經之一,現傅本九章算术是經魏刘徽(263年)和唐李淳風等人(656年,至今恰为1300年)注釋过的注釋本,它比較有系統地記載了我国古代数学的各个方面,本文的目的就是要尝試对其第8章方程中关於代数学方面,特别是关於“方程”解法方面的成就加以叙述。方程章第1問:“今有上禾三秉中禾二秉下禾一秉实三十九斗,上禾二秉中禾三秉下禾一秉实三十四斗,上禾一秉中禾二秉下禾三秉     

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.     

Computing curve intersection by homotopy methods

Intersection problems are fundamental in computational geometry, geometric modeling and design and manufacturing applications, and can be reduced to solving polynomial systems. This paper introduces two homotopy methods, i.e. polyhedral homotopy method and linear homotopy method, to compute the intersections of two plane rational parametric curves. Extensive numerical examples show that computing curve intersection by homotopy methods has better accuracy, efficiency and robustness than by the Ehrlich–Aberth iteration method. Finally, some other applications of homotopy methods are also presented.     

Computing curve intersection by homotopy methods

Intersection problems are fundamental in computational geometry, geometric modeling and design and manufacturing applications, and can be reduced to solving polynomial systems. This paper introduces two homotopy methods, i.e. polyhedral homotopy method and linear homotopy method, to compute the intersections of two plane rational parametric curves. Extensive numerical examples show that computing curve intersection by homotopy methods has better accuracy, efficiency and robustness than by the Ehrlich-Aberth iteration method. Finally, some other applications of homotopy methods are also presented.     

CATEGORICAL HOMOTOPY

We present a homotopy theory of small categories. In a work of this nature there is a need to give a theory which is clear and which shows the methods of work in this field. It is also necessary to prove theorems which place the theory within the general framework of homotopy, i.e. particularly to liaise with the homotopy of topological spaces and with abstract homotopy theories. Firstly we define the important notion of finite functor on which the theory is based. Next we introduce a type of fibred category fitting to the work on homotopy. After having studied the paths and loops of a category, we consider homotopy between functors. Finally, we demonstrate the possibility of obtaining homotopy groups before taking into consideration the relations between categorical and topological homotopy.     

Application of homotopy perturbation methods for solving systems of linear equations

In this paper, homotopy perturbation methods (HPMs) are applied to obtain the solution of linear systems, and conditions are deduced to check the convergence of the homotopy series. Moreover, we have adapted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix. The numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius.     

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

In this paper, we obtain approximate‐analytical solutions of a cancer chemotherapy effect model involving fractional derivatives with exponential kernel and with general Mittag‐Leffler function. Laplace homotopy perturbation method and the modified homotopy analysis transform method were applied. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on homotopy polynomial. The cancer chemotherapy effect equations are solved numerically and analytically using the aforesaid methods. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new fractional‐order derivatives with exponential decay law and with general Mittag‐Leffler law.     

求解混合三角多项式方程组的直接GBQ方法

许多科学与工程领域,我们经常需要求混合三角多项式方程组的全部解.一般来说,混合三角多项式方程组可以通过变量替换及增加二次多项式转化为多项式方程组,进而利用数值方法进行求解,但这种转化会增大问题的规模从而增加计算量.在本文中,我们不将问题转化,考虑利用直接同伦方法求解,并给出基于GBQ方法构造的初始方程组及同伦定理的证明.数值实验结果表明我们构造的直接同伦方法较已有的直接同伦方法更加有效.     

Efficient computational algorithms for solving one class of fractional boundary value problems

In this paper, we introduce a modification of He’s variational iteration, homotopy analysis and optimal homotopy analysis methods for solving fractional boundary value problems. It is illustrated that the proposed methods are powerful fast numerical tools to find accurate solutions. It is illustrated that efficiency of these methods is based on proper choosing of initial guess.     

Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein–Gordon–Zakharov equations in plasma physics

In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein–Gordon–Zakharov equations. We consider fractional coupled Klein–Gordon–Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein–Gordon–Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order α ∈ (1, 2], β ∈ (1, 2]. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.     

On the geometry of paths generated by PL homotopy methods

PL homotopy methods are effective numerical methods for highly nonlinear problems. It is widely believed that the feasibility of a PL homotopy method depends on the nondegeneracy condition that the zero set (or the fixed point set in the case of finding fixed points instead of zeroes) of the PL approximation of the homotopy does not intersect the triangulation's skeletons of co-dimensions two and above. This paper shows that, although the sections of the PL approximation's zero set tracked by the PL homotopy method are of dimension one (while other sections may have higher dimensions), the paths generated by the pivoting method are potentially and essentially of dimension two. It makes pathcrossing a safe thing. Thus, this paper first sets up the without exception feasibility of PL homotopy methods geometrically.This work is supported in part by the Foundation of Zhongshan University Advanced Research Centre.